Bicycle Repair/Dialing in 21 speed bike

First, thanks for reading my question. I'm a general, all around handyman/mechanic sort and while I have ridden road and mountain bikes, my expertise is in BMX. Me and my son built him, from scavenged parts, a 26" downhill type bike and I've got everything dialed in nice except the deraileurs. It has grip shfters, which turn smoothly and index well, but having been through both the front and rear over and over again, I'm out of ideas. I can either get the rear one to start in 1 and go all the way to 6, but won't go to 7, or I can dial it in starting at 7, but then it won't go all the way back to 1. The front should be simpler, but I can either get it to rub against the chain in 2, then jump to 3, or start at three but not go back to one. I hope you can offer a good procedure for me to follow to get this worked out! I look forward to your response!

Answer Setting up a shifter system comes with a number of pre-requisites: correct wheel/frame alignement; correct derailer/frame alignment; proper setting of the derailers' respective limit screws, etc.
Presuming you have met these pre-conditons, it's best to start the index-adjustment procedure with both derailers/shifters at their "normal" or relaxed positions. For the rear derailer system, this means setting the shifter at the "7" or "H". For the front, the shifter should be set at "1" or "L".
Next, confirm that each derailer is in fact in its relaxed or normal position: the rear derailer guide(upper) pulley, visible through the chain, should be directly in line with the "7" or smallest rear cog and the front derailer should be somewhat centered over the smallest chain ring. Deviation at this point usually implies: excessive cable tension; incorrect limit screw settings.
Starting with the rear system, begin by making the downshift from "7" to "6", and gauge how the rear derailer responds: if, while pedaling, the derailer hestitates and "clatters" before shifting, it's a sign of insufficient cable tension; a corresponding reluctance or delay in the subsequent upshift from 6 to 7 would suggest excessive cable tension. Adjust accordingly.
Once the rear derailer can be successfully shifted to the 1 or "L" or largest, innermost cog, it's time to set the front derailer indexing. Leave or set the chain in the L position at the rear, you can now more correctly evaluate the front derailer's alignment, "L" limit screw settings, cable tension, etc. A properly set front derailer will have, at this point, the inside edge of it's inner cage plate parallel to and just touching the side plate(s) of the chain, with the chain on the innermost or L chain ring (there will be slight chain rub at this point, under no load. It will disappear under load.). If this is not the case, it's a sign of incorrect "L" limit screw settings, excessive cable tension or a combination of both.
The next step is, while pedaling at an effective cadence, move the front derailer's shifter to the next highest index position: on a triple ring crank-set, it'll be the 2 setting. Correct cable tension should deliver the chain to the subsequent Middle ring in 1/4 to 1/2 of a turn of the crank. Any hesitation or delay is indicative of insufficient cable tension. Adjust accordingly. A correctly tensioned (or "indexed") cable system should again position the front derailer's inside edge of the inner cage-plate just touching the inside edges of the chain's plates(There will be some chain rub under no load condition; this will disappear under load). This is only the case with the chain in the 1 or "L" cog position in the rear.
The final front shift system check is to confirm the upshift to the outermost "H" or 3rd chain ring. Any hesitation at this point suggests incorrect "H" limit screw settings (too tight) or derailer/crank alignment issues.
Of course, all of these steps presuppose that all the related components are fully compatible; the fact that you are using "scavenged" parts has some potential drawbacks. Good luck!

Expertise

I can answer most inquiries with respect to bicycle assembly, repair, tuning and maintenance.
I am less than inclined to speak on such questions as "What kind of bike should I buy?", &/or "What is this bike worth?", etc. Any answers to these types of questions may correlate poorly to an acceptable "fact/sarcasm" ratio.